Further notes on the treatment of space-charge in CPO2DS

In CPO2D a cylindrical system has electrodes that have rotational symmetry around the axis, but the individual rays can have any symmetry or none.

But in the space-charge CPO2DS program a different assumption is made.

As a simple example, consider a parallel beam that is centered on the axis. It could be defined by 2 lines at its lower and upper edges. To calculate the space-charge field of this beam it would obviously be assumed that the beam is solid. In fact the lower line is not strictly needed, since the system has cylindrical symmetry around the axis. Also these 'lines' are not rays, but merely define the solid beam.

The treatment in CPO2DS is similar to this, but more sophisticated. The 'lines' do not enclose solid volumes but instead they represent sheets of current, with cylindrical symmetry. The 'lines' are essentially cuts through the sheet, and in principle the line through the lower cut is not needed. In CPO2DS these 'lines' are referred to as 'rays'. Usually only the upper 'ray' is defined, but there is an option to display the lower ray after the calculation has finished.

So in CPO2DS the space-charge is symmetric around the axis, which increases the speed of the calculations. When a ray hits the axis there is a choice of letting it cross the axis or reflecting it at the axis. We have chosen the first option because this is what happens physically for a single isolated real particle. Then there are two conical sheets, the end points of which meet on the axis at the crossing point.

Of course, a different assumption could have been made in CPO2DS. Instead of assuming that all the space-charge is symmetric around the axis it could have been assumed that the rays represent the paths of individual particles, which is the assumption made in CPO3DS. But then the advantage of a faster calculation would have been lost. So CPO3DS has to be used for systems that have cylindrical symmetry for the electrodes but not for the rays (and here there are the 3 symmetry planes x=0, y=0, and x=y to speed up the calculation).